I finally fucking did it! What made the difference was: using pale nails, a lot, mostly for flying shrimps; ascendant's grip for when they remove the floor and you have to shoot at flying enemies; reaper's crest; dumping tacks where the alitas spawns in the third gauntlet. I didn't kill Khann on the very first attempt right after the gauntlet, but I did it on the second.
Sidenote, I decided to get pale nails for this fight so I went up the cradle, but once there I told myself "now that we are here let's just finish the Mr Mushroom thing" only to discover the Path of Regular Discomfort. Where I got stuck once again, until I swallowed my pride and went back to buy the ascendant's grip. Which turned out to be a life saver for Khann.
Side sidenote: reaper's crest is amazing against the pinstress. It's like, made specifically for that fight.
Side side sidenote: I imagine you are not that interested in my quest. But I just wanted to write this somewhere.